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We study tail asymptotics of the stationary distribution for the -type Markov chain with finitely many background states. Decay rate in the logarithmic sense is identified under a number of conditions on the transition probabilities. The results are applied to the queue with vacations. The relationship between vacation time and the decay rate of the queue length distribution is investigated.

Logarithmic Asymptotics for the GI/G/1-type Markov Chains and their Applications to the BMAP/G/1 Queue with Vacations